"Understandable." He can see why they would want to -- no one ever likes to live in the shadows of somebody else, or so he's been told before, anyway. "That's because he knows me better as someone else."
It's a risk if he reveals his secret to her, he realizes. But he also might need allies in this place. And he failed Clark, that much he knows, maybe he could help his cousin instead. It won't change anything, it won't fix the mistakes he's made in the past, but it's something.
"Batman."
Of course, she might not believe him. He doesn't have his suit or anything else to back up the claim here.